# Spahn directed object' (Rev #6, changes)

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###### Definition (interval object)

Let $C$ be a closed monoidal (V-enriched for some $V$) homotopical category, let $*$ denote the tensor unit of $C$. Then the cospan category (same objects and cospans as morphisms) is $V$-enriched, too.

An interval object is defined to be a cospan $*\stackrel{a}{\rightarrow}I\stackrel{b}{\leftarrow}*$.

The pushout $I^{\coprod_2}$ of this diagram satisfies ${}_*[ I,I^{\coprod_2}]_*\simeq$ is contractible (see co-span for this notation).

###### Definition (directed object)

Let $V$ be a monoidal category, let $C$ be $V$-enriched, closed monoidal homotopical category, let $*$ denote the tensor unit of $C$ which we assume to be the terminal object, let $*\stackrel{0}{\to} I\stackrel{1}{\leftarrow}$ denote the interval object of $C$, let $X$ be a pointed object of $C$. Let $D$ be the functor bifunctor D:C\to D:C\times C\to V,  X\mapsto (X,I)\mapsto [I,X].

A direction in $C$ is defined to be a subfunctor $d$ of  D D(I,-) preserving the terminal object and satisfying the properties below. In this case $d X$ is called a direction for $X$. A global element of $dX$ is called a $d$-directed path in $X$ and their collection we denote by $ddp(X)$ . the The properties are:

(1) The $D$ image of every map $I\to *\to X$ factoring over the point is in $ddp(X)$.

(2)  ddp(X) f,g\in dX is are closed called under to the be tensor a product. For$\alpha,\beta\in ddp(X)$composable pair , their if pushout \alpha\otimes \overline \beta f\circ 0=\overline g\circ 1 where the overlined letters denote the adjoints under the hom adjunction. The composition of a composable pair is called defined their bycomposition$f\circ g:=\underline{\overline{f}\bullet\overline{g}}$ . Then the condition is that$ddp(X)$ is closed under composition.

A (3) Ifdirected object$\tau\in hom (I,I)$ is and defined to be a pair {}_d f\in X:=(X,dX) ddp(X) consisting then of an object X \underline{\overline{f}\circ \tau}\in dX of where the underlined term denotes the adjoint (in the other direction) under the hom adjunction.$C$ and a direction $dX$ for $X$.

A (4)morphism of directed objects$ddp(X)$ is a functor$f:(X,dX)\to (Y,dy)$ is defined to be a pair $(f,df)$ making

$\array{ X& \stackrel{f}{\to}& Y \\ \downarrow^d&&\downarrow^d \\ dX& \stackrel{df}{\to}& d Y }$

A directed object is defined to be a pair ${}_d X:=(X,dX)$ consisting of an object $X$ of $C$ and a direction $dX$ for $X$.

A morphism of directed objects $f:(X,dX)\to (Y,dY)$ is defined to be a pair $(f,df)$ making

###### Remark

Let $C$ be $V$-enriched, closed monoidal homotopical category, let $I$ denote the tensor unit of $C$.

###### Definition

A directed-path-space objects is defined.

Revision on November 6, 2012 at 00:41:31 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.